Given a quantum channel \(U_{AB}:\mathcal {H}_A\otimes \mathcal {H}_B\rightarrow \mathcal {H}_C\otimes \mathcal {H}_D\)  and a state \({\psi }_A\in \mathcal {H}_A\) , we consider the following quantum state [1]}:
\((I_{C}\otimes V_{DB^{\prime }})(U_{AB}\otimes I_{B^{\prime }})({\psi }_A\otimes {\text{EPR}}_{BB^{\prime }})={\begin{array}{c}\vbox {\vspace{1.19998pt}\hbox{\hspace{0.79999pt}\includegraphics [scale=0.4]{fig_V_HPstate_psi_v2}}}\end{array}}.\) 
